The Stacks project

Lemma 36.27.4. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a flat proper morphism of schemes. Let $E \in D(\mathcal{O}_ X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ S)$.

Proof. We claim that Lemma 36.27.1 applies. Conditions (1) and (2) are immediate. Condition (3) is local on $X$. Thus we may assume $X$ and $S$ affine and $E$ represented by a strictly perfect complex of $\mathcal{O}_ X$-modules. Since $\mathcal{O}_ X$ is flat as a sheaf of $f^{-1}\mathcal{O}_ S$-modules we find that condition (3) is satisfied. $\square$

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